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27 Questions

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12	Finding $\displaystyle \lim_{n \to \infty} \int_0^\infty \frac{e^{-x}\cos{x}}{nx^2 + \frac{1}{n}}dx$
11	Determine $\displaystyle \lim_{n \to \infty}{{n} \choose {\frac{n}{2}}}\frac{1}{2^n}$, where each $n$ is even
8	Give an example of two closed sets $A, B \subseteq \mathbb{R}$ such that the set $A + B $ is not closed [duplicate]
5	Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$
5	Show that, given spherically symmetric initial data, a solution to the heat equation is spherically symmetric

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472 Reputation

+10	If we anti-differentiate the Laurent series $ \sum_{n=2}^{\infty} a_n(z - a)^{-n}$, how is the radius of convergence affected?
+5	Finding $\displaystyle \lim_{n \to \infty} \int_0^\infty \frac{e^{-x}\cos{x}}{nx^2 + \frac{1}{n}}dx$
+15	Determine the limit of $\frac{1}{n}\sum_{i =2}^n \frac{1}{\ln i}$ as $n \to \infty$
+5	Determine the limit of the expression $\frac {\sum_{k=1}^n \frac{1}{k} - (1-\frac{2}{n})\ln n}{\ln n}$ as $n \to \infty$

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4 Answers

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1	If we anti-differentiate the Laurent series $ \sum_{n=2}^{\infty} a_n(z - a)^{-n}$, how is the radius of convergence affected?
1	Show that the pointwise limit of monotone increasing harmonic functions is harmonic
0	Show that analytic function $f$ cannot be conformal when $f'(z) = 0$
0	For measurable $f_n : X \to [0, \infty)$ show that $\sum_{n=1}^\infty f_n < \infty$ almost everywhere

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1 real-analysis × 8	0 binomial-coefficients × 2
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